Monolayers of Globular Proteins on the Air/WaterInterface: Applicability of the Volmer Equation of State

Theodor D. Gurkov,*,† Stoyan C. Russev,‡ Krassimir D. Danov,†Ivan B. Ivanov,† and Bruce Campbell§

Laboratory of Chemical Physics & Engineering, Faculty of Chemistry, University of Sofia,James Bourchier Avenue 1, Sofia 1164, Bulgaria, Department of Solid State Physics, Facultyof Physics, University of Sofia, James Bourchier Avenue 5, Sofia 1164, Bulgaria, and Kraft

Foods, Incorporated, Technology Center, 801 Waukegan Road, Glenview, Illinois 60025

Received February 13, 2003. In Final Form: June 13, 2003

We performed simultaneous measurements of the instantaneous values of the surface pressure versustime, Π(t) (by the Wilhelmy plate method), and the adsorption versus time, Γ(t) (by ellipsometry), foraqueous solutions of a globular protein (â-lactoglobulin, BLG). The resulting dependence Π(Γ) was foundto be well described by the Volmer equation of state (when Γ e 1.6 mg/m2, a value corresponding to analmost complete monolayer), for all times and bulk concentrations. The excluded area per molecule, R,turned out to be twice as large as the maximum cross-sectional area of the molecule, ω, in accordance withthe theoretical considerations. We processed in the same way available literature data for various proteins(BLG, R-lactalbumin, bovine serum albumin), both for equilibrium and for nonequilibrium adsorbed layers,as well as for spread layers. In all cases, the experimental dependencies Π(Γ) were fitted well by the Volmerequation; the excluded area either was almost exactly twice the maximum cross-sectional area (for sphericalmolecules) or could be interpreted in a similar way (for nonspherical molecules), by means of qualitativelyequivalent reasoning. These results have led us to the following conclusions for the studied globularproteins: (i) The surface state depends only on the instantaneous adsorption, Γ, regardless of how it wasreached. (ii) The Volmer equation is obeyed for surface coverages close to or lower than the monomolecularadsorption. (iii) No denaturation occurs during the adsorption process.

1. Introduction

The adsorption of globular proteins on liquid interfacesis a widely studied phenomenon, due to its importance forthe stabilization of food dispersions (dressings and sauces,mayonnaise, ice cream, etc.). The complexity of the proteinadsorption comes from the fact that the molecules on thesurface usually undergo substantial conformationalchanges, unfolding and partial denaturation.1 The latterchanges are driven by the free energy gain when hydro-phobic moieties from the molecule enter into the hydro-phobic phase (oil or air). The extent of the configurationalrearrangement is often dependent on the layer density:2at lower surface concentrations the globular proteinmolecules tend to unfold more, and in tightly packed layersthe scope for reorganization is rather limited.3 Typically,these processes of partial denaturation are slow (withcharacteristic times of the order of several hours4). Theyaffect the macroscopic properties of the layer, such as thesurface pressure. On the other hand, for relatively shortinterfacial aging (e.g., up to about an hour in the case oflysozyme4) the layer state can be understood with thepremise that the protein molecules have not succeeded tounfold. This of course depends on the particular protein,the bulk concentration, and so forth; in the present work

we analyze several cases in which the layer behavior isnot affected by surface denaturation.

Information about the physical state of the proteinmolecules on the fluid interface can be obtained frommeasurements of the surface pressure, Π (the lowering ofthe interfacial tension due to the layer), and the adsorbedprotein amount per unit area, Γ. Different types of Π(Γ)relation, that is, surface equations of state, have beenproposed and tested in the literature. In refs 5-8, anequation of the Szyszkowski-Langmuir type was adopted,Π ∼ ln(1 - ωΓ), where ω has the meaning of (average)partial molecular area. Statistical thermodynamical con-siderations9 confirm that the parameter ω can be inter-preted as the area physically occupied by one molecule.The authors of refs 5-8 assumed that protein moleculeson the interface, in equilibrium with the bulk solution,could exist in a number of states with different molarareas. In that model, the main factor determining thestate of the adsorbed molecules was the value of Π: at lowsurface pressure the molecular states with larger size werestrongly favored over those with small size. The latterfact was interpreted as deeper denaturation.8 Two-dimensional aggregation of the protein in the surface layerwas considered in ref 5. The influence of the electrostaticinteractions between charged protein molecules on theair/water (A/W) boundary was investigated in refs 5-8,in the frames of the Gouy-Chapman theory. The multiple-

* Corresponding author. E-mail: tg@lcpe.uni-sofia.bg.† Laboratory of Chemical Physics & Engineering, Faculty of

Chemistry, University of Sofia.‡ Department of Solid State Physics, Faculty of Physics, Uni-

versity of Sofia.§ Kraft Foods, Inc.(1) Dickinson, E. J. Chem. Soc., Faraday Trans. 1998, 94, 1657.(2) Uraizee, F.; Narsimhan, G. J. Colloid Interface Sci. 1991, 146,

169.(3) Dickinson, E. J. Dairy Sci. 1997, 80, 2607.(4) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70,

403.

(5) Fainerman, V. B.; Miller, R. Langmuir 1999, 15, 1812.(6) Fainerman, V. B.; Miller, R. In Proteins at Liquid Interfaces;

Mobius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1998; p 51.(7) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Colloids

Surf., A 1998, 143, 141.(8) Makievski, A. V.; Fainerman, V. B.; Bree, M.; Wustneck, R.;

Kragel, J.; Miller, R. J. Phys. Chem. B 1998, 102, 417.(9) Hill, T. L. An Introduction to Statistical Thermodynamics;

Addison-Wesley: Reading, MA, 1960.

7362 Langmuir 2003, 19, 7362-7369

10.1021/la034250f CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 07/30/2003

state model incorporating electrostatics was comparedwith experimental data for the dependence of the surfacepressure, Π, on the bulk protein concentration, cb , forbovine serum albumin (BSA),5 â-casein,6,7 and humanserum albumin (HSA).6-8 Fits (with three or four adjust-able parameters) showed good agreement between theoryand experiment, for Π up to 15-25 mN/m (refs 5-8).

Serrien et al.10 introduced a model with a first-ordersurface chemical reaction between native and unfoldedforms of the protein molecule (with only the native formbeing directly exchangeable with the bulk solution).Reorientation was also included as a possible change. Forboth BSA and casein, two reaction mechanisms were foundto be operative on the interface, with characteristic timesof ca. 120 and 3000 s for BSA and ca. 12 and 300 s forcasein, respectively.10 Another model surface equation ofstate for globular proteins on the A/W boundary2 tookinto account the statistics of the adsorbed segments (atdifferent degrees of unfolding), the mixing enthalpy withinthe interfacial layer (i.e., the segment-solvent interac-tions), and the electric double layer free energy. Good four-parameter fits with that model were reported for theexperimental Π(Γ) relations for BSA and lysozyme,2including the region of constant Π at high Γ.

In a recent work, Meinders and colleagues11 usedanother equation of state:

Although this equation was originally derived for a layerof hard disks,12 Meinders and colleagues11 applied it toviscoelastic disks by assuming that ω depends on thesurface coverage. Thus, the authors of ref 11 developeda four-parameter model. They interpreted the Π(Γ) datafor spread layers of â-lactoglobulin (BLG), R-lactalbumin,BSA, and â-casein in an interval for Γ up to rather highvalues (4-5 mg/m2).

In contrast to those complicated cases, in whichdenaturation of the protein had been developing with time,some systems containing globular proteins were found toexhibit simpler behavior under certain conditions. (Suchconditions include not very long interfacial aging, so thatsignificant unfolding would not take place.) Sufficientlydiluted layers should obey the 2D ideal gas law, or lawsassuming that the protein molecules are nonpenetrableobjects occupying a constant surface.13 Indeed, at lowvalues of Π and Γ the spread BLG layers investigated inref 11 had constant molecular area, ω. As another example,in refs 13 and 14 the experimental Π(Γ) dependence forspread BLG layers on concentrated salt, at surfacepressures below ∼0.8 mN/m, was described very well bythe equation of Volmer:

This equation of state, valid for hard disks,9,15 has onlyone parameter, R, which is often called “excluded area per

molecule”; kT is the thermal energy, and the adsorptionΓ is measured in number of molecules per cm2. The valueof R reported in ref 13 (with the data taken from ref 14)is 74.3 nm2 for BLG; this is far greater compared to whatfollows from other data (see section 4.1 below). Theprobable reason for this discrepancy is the difference inthe composition of the aqueous subphase: in ref 14 it was35 wt % (NH4)2SO4. The validity of eq 2 was confirmedalso for spread egg albumin on concentrated salt (35 wt% (NH4)2SO4), at Π e 0.4 mN/m (ref 16).

In the present work, we have studied layers of BLGformedby adsorption ontheair/water interface (asopposedto the spread layers in the papers cited above), measuringΠ and Γ independently, as functions of time. It isdemonstrated that the instantaneous values of the surfacepressure, Π(t), and the adsorption, Γ(t), obey eq 2 whenthe adsorption is below (or close to) that for a completemonolayer, irrespective of the time and the bulk concen-tration in the solution. We discuss also literature data forBLG from different sources. These data (for equilibriumadsorbed or spread layers, or for adsorbed layers underdynamic conditions) turn out to comply with eq 2, and thevalue of the parameter R coincides with that obtainedfrom our results (see section 4.1 below): R ) 19-20 nm2.The Π(Γ) relations for diluted layers of R-lactalbumin andBSA, measured by other authors, are also found to be ingood agreement with eq 2 and give reasonable values forR (section 4.2). Therefore, it can be concluded that allinvestigated proteins do not undergo denaturation andthe molecules do not interact appreciably when the layerson the air/water surface are relatively diluted (withcoverage lower than or close to a monolayer).

2. Experimental Section

2.1. The Setup. The adsorbed protein amount was measuredby ellipsometry, using a “rotating analyzer” apparatus (with afixed angle of incidence of 50°).17,18 The scheme of the ellipsometeris shown in Figure 1a. The two ellipsometric parameters Ψ and∆, which characterize the protein layer, were determined asfunctions of time. An automatic Wilhelmy balance with computerdata acquisition was used to measure the surface pressure. Bothdata sets (surface pressure and ellipsometric angles) weresimultaneously recorded every second.

2.2. The Measuring Cell. The experiments were performedin the measuring cell shown in Figure 1b. A glass semicylinder(with diameter 6 cm and length 12 cm) was filled up with aqueoussolution. A barrier made from poly(tetrafluoroethylene) (PTFE,Teflon), protruding less than 1 mm inside the solution, allowedus to clean up the surface and remove the adsorbed layer. At theend of the cylinder, a Wilhelmy plate was mounted. Theellipsometric beam spot was situated more than 2 cm away fromthe measuring plate and from the cylinder walls, to avoid anychange of the angle of incidence due to surface curvature. Thecell was covered with a glass plate (except in the region of thelaser beam and the Wilhelmy plate), to protect the surface fromcontamination and airflow.

2.3. Materials and Experimental Procedure. Proteinsolutions at three different bulk concentrations cb, 0.01, 0.005,and 0.0005 wt %, containing 0.1 g/L sodium azide (NaN3 ,antibacterial agent), were prepared with deionized water froma Milli-Q system (Millipore); â-lactoglobulin from bovine milk(mixture of A and B variants, catalog no. L-0130) was purchasedfrom Sigma. The pH was adjusted to 5.2-5.4 by HCl. Theisoelectric point of BLG is at pH ≈ 5.2, so in our system theprotein molecules were essentially uncharged. All solutions wereused within 2 h after their preparation.

(10) Serrien, G.; Geeraerts, G.; Ghosh, L.; Joos, P. Colloids Surf.1992, 68, 219.

(11) Meinders, M.; van Aken, G.; Wierenga, P.; Martin, A.; de Jongh,H. In 14th Symposium “Surfactants in Solution”, Barcelona, Spain,June 9-14, 2002; Oral contribution no. O.082.

(12) Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J. Chem. Phys.1965, 43, 774.

(13) Douillard, R.; Aguie-Beghin, V. Colloids Surf., A 1999, 149, 285.(14) Bull, H. B. J. Am. Chem. Soc. 1945, 67, 8.(15) Israelachvili, J. N. Intermolecular and Surface Forces; Academic

Press: London, 1992.

(16) Bull, H. B. J. Am. Chem. Soc. 1945, 67, 4.(17) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized

Light; North-Holland: Amsterdam, 1977.(18) Russev, S. C.; Arguirov, T. V. Rev. Sci. Instrum. 1999, 70, 3077.

Π ) kTΓ(1 - ωΓ)2

(1)

Π(1Γ - R) ) kT (2)

Monolayers of Globular Proteins Langmuir, Vol. 19, No. 18, 2003 7363

The experimental cell and the glassware were dipped infresh sulfochromic acid (solution of potassium dichromate insulfuric acid) for 10 min; then they were put in distilled waterfor 20 min, rinsed with deionized water (from Millipore), afterthat soaked in Millipore water for 20 min, and subsequentlyrinsed with Millipore water. The cell was then dried and placedon the sample holder. We tested the cleanness of the setup wallsand the pure water used for the solutions in the followingway: The vessel was filled up with Millipore water and left forseveral hours to allow emerging of any surface-active contami-nation. Meanwhile, the water surface was monitored ellipso-metrically. Afterward, the surface was highly compressed by the

barrier, and if no change in the ellipsometric signal occurred,we assumed that the system was clean. The pure water wasthen sucked out and replaced by protein solution. After that,the surface of the protein solution was cleaned by sweeping withthe barrier, and immediately the tensiometer plate was put incontact with the interface. From that moment on, the measur-ing devices were simultaneously operating. The data comingfrom both apparatuses (ellipsometer and tensiometer) wererecorded and stored every second for 30-40 min after the surfacecleaning. Next, the surface of the protein solution was cleanedagain by the barrier and a new experiment (called a “run”) wasperformed.

Figure 1. Scheme of the experimental setup. (a) The ellipsometer, equipped with rotating analyzer unit, which allows dataacquisition at every second. (b) The experimental cell, in which ellipsometry and surface tension measurements are performedsimultaneously. There is a barrier for cleaning the air/water boundary.

7364 Langmuir, Vol. 19, No. 18, 2003 Gurkov et al.

3. Data ProcessingIn ellipsometry, one determines the change of the

polarization state of the incident light after its interactionwith the sample. This change is described by twoparameters, the ellipsometric angles ψ and ∆ (or equiva-lently, by the complex ellipsometric ratio F):17

The two measured quantities, ψ and ∆, are connectedwith the optical parameters of the sample by the relation

where Rp and Rs are the generalized Fresnel coefficientsfor p- (parallel) and s- (perpendicular) polarization,respectively. For a given model of the optical system, Rpand Rs are usually known functions of the refractive indicesand the layer(s’) thickness. Thus, eqs 3 and 4 can be usedto find two unknown system parameters from a singleellipsometric measurement (i.e., from a couple ψ, ∆). Inthe case of the system air/(surface protein layer)/bulksolution, these two quantities are the thickness, d, andthe refractive index, n, of the layer.

It is well-known17 that for very thin layers (in com-parison to the used wavelength) only one of the measuredellipsometric angles (viz., ∆) is sensitive to the change ofthe layer thickness and/or refractive index. The otherangle, ψ, changes only very slightly from its value for aclean surface. This means that we have only one usefulmeasured quantity, and consequently, only one physicalparameter of the layer can be found. On the other hand,the optical model of the system presumes that theinterfacial layer isahomogeneousvolumephasewith finitethickness and plane-parallel surfaces (air/layer and layer/water); for such a layer the unknown optical parametersare two: the thickness and the refractive index. Thisproblem is usually overcome by using additional informa-tion (e.g., dn/dc data, where c represents the concentrationof the solute per unit volume of the layer phase).

Since the protein layers are very thin (thickness/wavelength ≡d/λ , 1), we can use the following expressionfor the change of ∆ (which is valid up to the first-orderterm in d/λ, cf. refs 17, 19, 20):

where

Here δ∆ is the difference in ∆ between the interfacecarrying a layer and the bare interface (to which ∆h refers),æ is the incidence angle, n0 , n1x, and n2 are the refractiveindexes of the medium (upper phase), the layer, and thesubstrate, respectively, and d1x is the thickness of the layer.

The two unknowns, n1x and d1x, cannot be separatelydetermined solely from eqs 5, but the so-called “dn/dc

approximation” can be used to provide one additionalrelation:

where dn/dc ) 0.180 cm3/g is a typical value for pro-teins,21,22 n2 ) 1.332 is the refractive index of water, andc1x is the protein concentration in the layer (in units ofg/cm3). Further, the surface coverage, Γ1x, is related to theconcentration c1x:

Equations 5-7 lead to a relationship between thesurface coverage (adsorption), Γ1x, and the ellipsometricangle ∆:

with

Although the coefficient k in eq 8 depends on the layerrefractive index n1x, this dependence is relatively weak ifn1x does not change considerably. (The entire expectedrange of the refractive index n1x during the layer buildupis 1.332 < n1x < 1.5.) Therefore, using constant k as anapproximation would give sufficient accuracy for Γ1x inthin layers. The coefficient k was calculated according toeq 8a, for the particular system and angle of incidenceunder consideration (the result was k ≈ 0.6 mg m-2 deg-1).

4. Results and Discussion4.1. Equation of State from Π-Γ Data for BLG.

The ellipsometrically measured time dependence of theadsorbed amount of BLG, at the three studied bulkconcentrations, cb, is presented in Figure 2. One noticesthat the adsorption process is initially rather fast; in thefirst seconds after starting the experiment Γ already

(19) Antippa, A. F.; Leblanc, R. M.; Ducharme, D. J. Opt. Soc. Am.A 1986, 3, 1794.

(20) den Engelsen, D.; de Koning, B. J. Chem. Soc., Faraday Trans.1 1974, 70, 1603.

(21) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70,415.

(22) Handbook of Biochemistry; Sober, H. A., Ed.; The ChemicalRubber Co.: Cleveland, 1968.

F ) ei∆ tan ψ (3)

F ) Rp/Rs (4)

δ∆ ) ∆ - ∆h )

4πλ

n0 sin æ tan æ

[n22 - n0

2][1 - (n0/n2)2 tan2 æ]

Fx ) AFx (5a)

Fx ) d1x(n1x2 +

n02n2

2

n1x2

- n02 - n2

2) (5b)

Figure 2. Ellipsometric results for the time dependence of theamount of BLG adsorbed at an air/water interface, at threebulk concentrations cb.

n1x ) n2 + dndc

c1x (6)

Γ1x ) c1xd1x (7)

Γ1x ) k(∆ - ∆h ) (8)

k )n1x

2

A(dn/dc)[n1x2 - n0

2](n1x + n2)(8a)

Monolayers of Globular Proteins Langmuir, Vol. 19, No. 18, 2003 7365

exceeds 1 mg/m2. The very beginning (the first severalseconds) of the adsorption kinetics is actually lost in ourmeasurement. We could not start exactly from time zerobecause initially the interface was perturbed by thecleaning barrier (Figure 1b), as well as by the insertionof theWilhelmyplate.Therefore, the threecurves inFigure2 may have different (and unknown) shifts along theabscissa, with magnitudes of the order of seconds. In ourinterpretation below, however, the time will be excludedto yield the Π(Γ) relation, so these time shifts will hardlyaffect the results for the equation of state which followfrom the Π(Γ) data.

From Figure 2, one sees that after the initial fastadsorption the process slows down and in a few (around5) minutes Γ reaches plateau values; it levels off atabout 1.55-1.60 mg/m2. This plateau adsorption is closeto what was found in ref 23 by means of neutronreflectivity: Γ ) 1.64 mg/m2 was measured for BLGmonolayers on the air/water boundary, at 10-3 and 10-2

wt % protein in the bulk.23

The time dependence of the surface pressure, Π(t),measured simultaneously with Γ(t), is shown in Figure 3.We observe that Π is a linear function of the logarithmof time, at all studied concentrations of BLG. A similartrend of Π ∼ ln(t) was reported by Beverung et al.,24 fordifferent globular proteins at the interface betweenheptane and water. It is not our purpose in the presentwork to explain the peculiar dependence Π ∼ ln(t). Wemention only that a barrier adsorption mechanism is likelyto hold, with dΠ ∼ e-const.Π dt (the latter relation gives Π∼ ln(t) directly). The exponent suggests existence of abarrier, probably connected with the jumping of moleculesfrom the subsurface onto the interface (see, e.g., ref 25).

Another specific feature of the curves in Figure 3 is thepresence of lag time. Especially at a low bulk concentration(5 × 10-4 wt %), the surface pressure is seen to remainzero while the adsorption has increased to reach valuesabove 1 mg/m2. This phenomenon with globular proteinsis known in the literature. Ybert and di Meglio26 discussedthe case of BSA on the water/air interface. It has been

suggested that during the lag time the adsorbed layer isdiluted and obeys the two-dimensional ideal gas law. Thelatter predicts very small changes of the surface tension(below 0.1 mN/m), which are not measurable.26 Thus,although there are adsorbed molecules on the interface,due to the lack of appreciable interactions the surfacepressure remains vanishingly small.

By eliminating the time, we combined our results fromFigures 2 and 3, for Γ(t) and Π(t) at different cb, todetermine thesurfaceequationof state, that is, therelationΠ(Γ). We tried to fit those data using the Langmuirisotherm with a single constant parameter, ω: Π ) -(kT/ω) ln(1 - ωΓ); however, the equation in this form turnedout to be inadequate. The attempt to utilize eq 1 was alsounsuccessful. The most appropriate equation for thepurpose of Π(Γ) fit was the Volmer isotherm, eq 2. Wepresent it as

and draw the corresponding plot in Figure 4. What isremarkable here is that the points of Π and Γ at differenttimes and bulk protein concentrations all lie on the samestraight line. This fact proves that the physical state ofthe surface layer is entirely determined by the instan-taneous value of the adsorption, Γ. There are no prehistoryor aging effects; at each moment of time the surfacepressure corresponds to the instantaneous Γ. Thus, forthe duration of the experiment the protein moleculesresiding on the interface do not undergo any noticeableconfigurational rearrangement and denaturation. Theexcluded area per molecule, R, determined from the linein Figure 4, is 19.3 nm2. The latter value will be discussedbelow in view of the molecular size of BLG.

The fact that eq 2 is the only suitable one-parameterequation is indicative for the physical behavior of theprotein molecules in the adsorption layer. The Volmer eq2 corresponds to nonlocalized adsorption of hard diskswhich do not interact with any long-range forces.9,15,27

Equation 2 can be regarded as a particular case of thetwo-dimensional van der Waals equation of state, withthe long-range interaction parameter equal to zero.15

Hence, one may infer that the adsorbed molecules of BLG(23) Atkinson, P. J.; Dickinson, E.; Horne, D. S.; Richardson, R. M.

J. Chem. Soc., Faraday Trans. 1995, 91, 2847.(24) Beverung, C. J.; Radke, C. J.; Blanch, H. W. Biophys. Chem.

1999, 81, 59.(25) MacRitchie, F.; Alexander, A. E. J. Colloid Sci. 1963, 18, 458.(26) Ybert, C.; di Meglio, J.-M. Langmuir 1998, 14, 471.

(27) Landau, L. D.; Lifshitz, E. M. Statistical Physics; PergamonPress: Oxford, 1980; Part I.

Figure 3. Results for the time dependence of the surfacepressure of adsorbed BLG layers (Π ) σ0 - σ, i.e., the decreaseof the surface tension, σ, with respect to that at the bare air/water interface).

Figure 4. Plot of the data from Figures 2 and 3 according tothe Volmer equation of state, eq 9. The adsorption, Γ, is takenin units of molecules/cm2.

ΠΓkT

) 1 + ΠRkT

(9)

7366 Langmuir, Vol. 19, No. 18, 2003 Gurkov et al.

behave as hard disks, if the surface coverage, Γ, is notgreater than the value corresponding to the monolayerstate (i.e., 1.64 mg/m2, ref 23). We have ellipsometric datashowing that at longer times, up to 10 h, nothing happensin the system with cb ) 0.01 wt % BLG; Γ remainspractically constant (the layer has reached saturation).In contrast, at higher bulk concentrations one observescomplicated phenomena connected with denaturation and/or multilayer formation.

We compare now our findings about the applicabilityof the Volmer equation with other authors’ data forâ-lactoglobulin. A graph with such data, plotted accordingto eq 9, is displayed in Figure 5. We have selected thefollowing three sets of experimental results (sets i and iipertain to adsorbed layers from bulk aqueous solutionswith concentration cb, and set iii to spread layers): (i)equilibrium isotherms for Π(cb) and Γ(cb), measured inseparate experiments (by axisymmetric drop shape analy-sis and by ellipsometry, respectively), from ref 28; (ii)dynamic Π(t) and Γ(t) data at cb ∼ 10-4 wt % BLG, obtainedsimultaneously in a Langmuir trough (by means of aWilhelmy plate and a radiotracer technique with 14C,respectively), from refs 29 and 30; (iii) equilibrium Π(Γ)isotherm for spread layers of BLG (at pH ) 5.6),compressed by a barrier in a trough, from ref 31.

Figure 5 demonstrates that all types of Π-Γ data, forequilibrium or dynamic adsorption, or with spread layers,satisfy one and the same surface equation of state: thatof Volmer (eqs 2 and 9), with the same value of theadjustable parameter, R. The latter is determined fromthe slope of the line in Figure 5. Let us denote 1/R withΓ∞; in the framework of the model eq 2, this is the maximumpossible adsorption, at which Π would diverge. Theliterature data collected in Figure 5 yield

The results from our measurements (Figure 4) give similar

values of the relevant parameters:

We have analyzed also the Π(Γ) data of Meinders et al.11

for spread BLG layers at Γ < 1.5 mg/m2: they complywith eq 9, with R ≈ 22 nm2 (which is close to R in eqs 10).Thus, there is a good agreement between the differenttypes of data with BLG, measured independently withdifferent methods and by different authors. The obtainedvalues of Γ∞ are close to the monolayer adsorption, 1.64mg/m2 (ref 23). We may conclude that irrespective of theconcrete procedure for preparation of the BLG layer onthe air/water interface, the protein molecules behave asnoninteracting hard disks if Γ e ∼1.65 mg/m2.

An attempt to rationalize the physical significance ofthe constant R can be made in view of the meaning of theparameters in the van der Waals and Volmer equationsof state. If the layer is diluted, so that virial expansion ofΠ with respect to the surface density, Γ, is possible, thenthe statistical derivation of the van der Waals equationsuggests that R is the excluded area per molecule.9,15,27

The two-dimensional equation of state for a layer ofmolecules with finite size is Π(a - aexc) ) kT, where a )1/Γ is the area per molecule, and the excluded area, aexc,is the area around each molecule which is inaccessible forother molecules (taken as a statistical average). If d is thediameter of a molecule (a hard disk), then d is theminimum possible center-to-center distance betweenapproaching molecules, see Figure 8a. The inaccessiblearea (shaded in Figure 8a) is equal to πd2, and the excludedarea per one molecule would be aexc ) πd2/2. The multiplier(1/2) takes into account the fact that the excluded areaeffect is due to the mutual interaction between themolecules, that is, each act of exclusion always involvestwo molecules, A and B. So, only half of the inaccessiblearea can be ascribed to a given molecule. A comprehensivediscussion and proof of these considerations can be foundin ref 9 (the chapter for the van der Waals equation ofstate), as well as in ref 27 and in section 6.3 of ref 15.

Now, the comparison of the “real gas” equation of state,Π(a - aexc) ) kT, with eq 2 yields R ) aexc ) πd2/2 ) 2ω,where ω ) πd2/4 is the disk area, that is, the area physicallyoccupied by one molecule. In this approach, only binaryinteractions between the particles are accounted for (letus recall that the layer is assumed to be diluted). In aconcentrated system of hard spheres, the excluded volumebecomes an extremely complicated function of the density(due to overlapping). The problem was explored theoret-ically in ref 32.

The BLG molecule in a bulk aqueous solution is a spherewith diameter 3.58 nm,29 which has a cross-sectional areaof 10 nm2. If the protein molecules do not undergosignificant configurational changes on the water/airboundary, then we would have ω ≈ 10 nm2 and R ) 2ω≈ 20 nm2, which compares well with the experimentalvalue of R ≈ 19-20 nm2. The fact that the surfacedimension coincides with the bulk dimension of themolecule suggests that substantial denaturation (ac-companied with changes in the molecular shape) is notvery likely to take place in the undersaturated layersconsidered above.

The relation R ) 2ω, with ω being the disk area, followsalso from eq 1 for diluted layers: at small ωΓ thedenominator of eq 1 becomes 1 - 2ωΓ, so eq 1 transformsinto the Volmer eq 2, with 2ω corresponding to the constant

(28) Miller, R.; Fainerman, V. B.; Makievski, A. V.; Grigoriev, D. O.;Wilde, P.; Kragel, J. In Food Emulsions and Foams; Dickinson, E.,Rodriguez Patino, J. M., Eds.; Royal Society of Chemistry: Cambridge,1999; p 207.

(29) Cornec, M.; Cho, D.; Narsimhan, G. J. Colloid Interface Sci.1999, 214, 129.

(30) Cornec, M.; Narsimhan, G. Langmuir 2000, 16, 1216.(31) Ivanova, M. G.; Verger, R.; Bois, A. G.; Panaiotov, I. Colloids

Surf. 1991, 54, 279. (32) Corti, D. S. J. Phys. Chem. B 2001, 105, 11772.

Figure 5. Literature data for BLG at the air/water interface,plotted according to eq 9. Set 1: Equilibrium isotherms forΠ(cb) and Γ(cb), from ref 28. Set 2: Dynamic Π(t) and Γ(t) dataat cb ∼ 10-4 wt % BLG, from refs 29 and 30. Set 3: EquilibriumΠ(Γ) isotherm for spread layers of BLG (at pH ) 5.6), from ref31.

R ) 18.6 nm2

Γ∞ ) 1/R ) 5.38 × 1012 cm-2 ) 1.65 mg/m2 (10a)

R ) 19.3 nm2

Γ∞ ) 1/R ) 5.19 × 1012 cm-2 ) 1.59 mg/m2 (10b)

Monolayers of Globular Proteins Langmuir, Vol. 19, No. 18, 2003 7367

R. However, as emphasized very clearly by Landau andLifshitz,27 the van der Waals equation should be regardedas an interpolation formula whose validity should bejudged from the comparison with the experiment. It shouldnot be considered to hold for low densities only (in gaseoussystems). Indeed, in our case the Volmer equation is foundto be valid also when the surface layer of BLG is not verydiluted.

4.2. Other Globular Proteins. The findings with BLGstimulated us to check if diluted layers (with surfacecoverage up to a monolayer) of other globular proteinswould also follow eq 2. Figure 6a shows literature data foradsorbed R-lactalbumin on the air/water interface, takenfrom ref 29. Dynamic Π(t) and Γ(t) values (measured byWilhelmy plate and by radioactivity of 14C-labeled protein,in a Langmuir trough), at cb ) 0.5 × 10-4, 1 × 10-4, and2×10-4 wt % protein, are used to extract the Π(Γ) relation.Obviously, the Volmer equation is satisfied, and the slopeof the straight line in Figure 6a gives

The size of the R-lactalbumin molecule in a bulk aqueoussolution is 2.3 × 3.7 × 3.2 nm (ref 29). The respectivecross-sectional areas, corresponding to different orienta-tions of the ellipsoid, are 6.7, 5.8, and 9.3 nm2. For thesake of clarity, the three possible orientations on a surface,with the respective cross-sections, are depicted in Figure6b. The areas are estimated according to the formula foran ellipse (π × 2.3 × 3.7/4 ) 6.7 nm2, etc.). The meanvalue of the three areas is 7.3 nm2. If we suppose that theprotein molecule behaves at the surface as a disk witheffective area ω ≈ 7.3 nm2, corresponding to the threepossible states in Figure 6b as an average, then we willobtain for the constant R ) 2ω a value of 14.6 nm2. The

latter estimate is quite close to the experimental value ofR from eq 11, R ) 14.9 nm2. As an alternative to thisinterpretation, there is one more possibility to explainthe experimental results. Indeed, if the protein moleculeis oriented with the side 2.3 × 3.7 nm sitting on theinterface (the first case in Figure 6b), then the calculatedvalue of the parameter R ) 2ω would be 2 × 6.7 ) 13.4nm2, which is again close to the experimental value of Rfrom eq 11, R ) 14.9 nm2. The other two molecularorientations then seem less probable. In any case, we canconclude that the relation R ) 2ω is confirmed, similarlyto what was found with BLG.

The structure of the R-lactalbumin molecule has beenconsidered as less stable in solution compared to BLGand more capable of denaturation.33 Nonetheless, theabove results suggest that surface denaturation of lac-talbumin (accompanied with substantial molecular shapechanges) does not take place, at least in the interval ofconcentrations and times where the data from Figure 6ahave been measured.

BSA is another globular protein whose properties onliquid interfaces have been studied extensively in theliterature. We verify the applicability of the Volmerequation (eqs 2 and 9) to equilibrium Π-Γ data for spreadmonolayers of BSA at pH ) 5.6 in a trough, taken fromref 31. In Figure 7, we see again a good agreement withthe Volmer formula; the adjustable parameter R is

The BSA molecule in a bulk aqueous solution hasapproximately cylindrical shape, with dimensions 4 × 4× 14 nm (ref 34). It has been proven that in relativelydiluted layers BSA lies in a side-on position at the liquidboundary.34 The latter conclusion was based on measure-mentsof the layer thicknessbyneutronreflection.34 Takingthe molecular size in the bulk state, cited above, we inferthat the corresponding cross-sectional area at the surfacewould be 56 nm2. The value of R from eq 12 is not far from(slightly larger than) twice the latter area (2ω ) 112 nm2,if ω ≈ 56 nm2); that is, the relation R ≈ 2ω may turn outto be fulfilled.

(33) Suttiprasit, P.; Krisdhasima, V.; McGuire, J. J. Colloid InterfaceSci. 1992, 154, 316. Suttiprasit, P.; McGuire, J. J. Colloid Interface Sci.1992, 154, 327.

(34) Lu, J. R.; Su, T. J.; Thomas, R. K. J. Colloid Interface Sci. 1999,213, 426.

Figure 6. (a) Literature data for adsorbed R-lactalbumin onthe air/water interface (dynamic Π(t), Γ(t) values, at cb ) 0.5× 10-4, 1 × 10-4, and 2 × 10-4 wt % protein), taken from ref29 and plotted according to eq 9. (b) Plane projections of differentorientations of the R-lactalbumin molecule. The three casescorrespond to dimensions of 2.3 × 3.7 nm, 2.3 × 3.2 nm, and3.2 × 3.7 nm, respectively.

R ) 14.9 nm2

Γ∞ ) 1/R ) 6.72 × 1012 cm-2 ) 1.56 mg/m2 (11)

Figure 7. Literature data for spread monolayers of BSA onthe air/water interface at pH ) 5.6 (equilibrium Π-Γ values),taken from ref 31 and plotted according to eq 9.

R ) 124.2 nm2

Γ∞ ) 1/R ) 8.05 × 1011 cm-2 ) 0.895 mg/m2 (12)

7368 Langmuir, Vol. 19, No. 18, 2003 Gurkov et al.

Of course, we are aware that the nonspherical shape ofthe (probably rotating) molecules could infringe theinterpretation of the constant R as being equal to 2ω, asdiscussed in section 4.1 above and illustrated in Figure8a for the case of disks. To clarify this point, we estimatethe excluded area for rectangular-shaped molecules,following the reasoning for disks. Figure 8b shows collisionof rectangular particles on a surface (with one particularpossibility for mutual orientation). The molecule B cantake different positions around the central molecule A,and the shaded area in Figure 8b is inaccessible for thecenter point of B. The corresponding excluded area permolecule is one-half of the inaccessible area, due tostatistical averaging (in just the same way as it was in thecase of disks, see section 4.1 above). Each of the particlesoccupies an area ω ) Ld; for BSA L ) 14 nm and d ) 4nm (the length and the cross-sectional diameter of thecylindrical molecule, respectively); ω ) 56 nm2. Thus, theinaccessible area around the central particle A in Figure8b will be (2L)(2d) ) 224 nm2, so the excluded area perone molecule is aexc ) 2Ld ) 112 nm2. There is one more

extreme possibility for mutual orientation of the twomolecules A and B upon collision (not shown as a picture),whose contribution to the excluded area can be calculatedin a similar manner, aexc ) 162 nm2. If all configurationshave equal probability, then the average excluded areawould amount to 137 nm2 (that is, 2.45ω, instead of 2ωfor disks). Comparing with eq 12, we observe that theexcluded area obtained from the experimental data, R )2.22ω (viz., 124.2 nm2 ) 2.22 × 56 nm2), is in reasonablygood agreement with the above assessment. As far as thelatter is implemented with ω corresponding to themolecular size in the bulk state, we can say that BSAunder these conditions is not likely to unfold and denatureat the air/water interface.

5. ConclusionsIn this work, we present experimental results for

simultaneously measured surface pressure (Π) and ad-sorbed amount (Γ) of â-lactoglobulin on the air/waterinterface. From the time-dependent Π(t) and Γ(t) atdifferent bulk concentrations, we eliminate the time andextract the Π(Γ) relation, which is well described by theVolmer equation of state. The layer obeys one and thesame equation of state irrespective of the surface age andthe concentration in the bulk, for values of Γ up to thosefor a complete monolayer (∼1.64 mg/m2). The latter factpoints to the absence of processes of denaturation andunfolding, aging and prehistory effects; the surfacepressure corresponding to a given Γ establishes instan-taneously. The Volmer equation suggests that the layerbehaves as a collection of noninteracting hard disks.

Our results are compared with literature data for BLG,for equilibrium and dynamic adsorption, and with datafor spread layers. In all cases, despite the differences inthe way of layer preparation, the Volmer equation issatisfied. There is one adjustable parameter, the excludedarea per molecule, R, whose value is determined from thedata fits. The result for R which follows from ourmeasurements (R ≈ 19.3 nm2) is in good agreement withthe outcome from the other authors’ data. If the proteinmolecule does not undergo substantial unfolding anddenaturation upon adsorption, then the area occupied byit on the A/W surface, ω, would be equal to about 10 nm2

(the cross-sectional area in the bulk solution state, whichis a sphere). Thus, we observe R ≈ 2ω, in accordance withthe statistical interpretation of the constant parameter,R, in the Volmer equation.

We discuss other two globular proteins: R-lactalbuminand BSA. Their layers on the air/water interface, undercertain conditions (low coverage and relatively shortsurface age), also obey the Volmer equation of state.Reasonable values for the excluded molecular area, R, areobtained from the fits of Π(Γ) data taken from literaturesources.

Acknowledgment. This work was financially sup-ported by Kraft Foods, Incorporated, Glenview, IL. Theauthors thank Dr. Marcel Meinders from the WageningenCentre for Food Sciences, The Netherlands, for sendinga preprint with experimental data for different proteins.

LA034250F

Figure 8. Sketches illustrating the excluded area of amolecule: (a) circular shape; (b) rectangular shape. The centralmolecule, A, occupies an area ω ) πd2/4 (a) or ω ) Ld (b) ona plane interface. A second molecule, B, comes to a closeapproach. With empty circles (a) or rectangles (b) we showdifferent possible positions of the second molecule, B, aroundthe central molecule A. The inaccessible area for the centerpoint of B is hatched in gray; the excluded area per moleculeis one-half of that area (due to statistical averaging).

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